Systems and methods for reducing peak to average cross-correlation for sequences designed by alternating projections

ABSTRACT

A method for using a numerical method to design reference signals for multiple input multiple output (MIMO) systems is described. An input multiple input multiple output signal is determined. A nearest tight frame to one or more given structured vectors is obtained. One or more structured vectors from the nearest tight frame are obtained. Orthogonal subsets are computed for a plurality of sequences, wherein each of the subsets is replaced with a matrix that comprises a function of one or more elements in each of the plurality of sequences and an identity matrix associated with the one or more elements. The one or more structured vectors are projected onto the space of circulant matrices. One or more classes of matrices associated with reference signals are outputted.

RELATED APPLICATIONS

This application is related to and claims priority from U.S. ProvisionalPatent Application Ser. No. 60/895,658 filed Mar. 19, 2007, for SYSTEMSAND METHODS FOR REDUCING PEAK TO AVERAGE CROSS-CORRELATION FOR SEQUENCESDESIGNED BY ALTERNATING PROJECTIONS, with inventor John M. Kowalski andHuaming Wu, which is incorporated herein by reference.

TECHNICAL FIELD

The present invention relates generally to wireless communications andwireless communications-related technology. More specifically, thepresent invention relates to systems and methods that reduce peak toaverage cross-correlation for sequences designed by alternatingprojections.

BACKGROUND

A wireless communication system typically includes a base station inwireless communication with a plurality of user devices (which may alsobe referred to as mobile stations, subscriber units, access terminals,etc.). The base station transmits data to the user devices over a radiofrequency (RF) communication channel. The term “downlink” refers totransmission from a base station to a user device, while the term“uplink” refers to transmission from a user device to a base station.

Orthogonal frequency division multiplexing (OFDM) is a modulation andmultiple-access technique whereby the transmission band of acommunication channel is divided into a number of equally spacedsub-bands. A sub-carrier carrying a portion of the user information istransmitted in each sub-band, and every sub-carrier is orthogonal withevery other sub-carrier. Sub-carriers are sometimes referred to as“tones.” OFDM enables the creation of a very flexible systemarchitecture that can be used efficiently for a wide range of services,including voice and data. OFDM is sometimes referred to as discretemulti-tone transmission (DMT).

The 3^(rd) Generation Partnership Project (3GPP) is a collaboration ofstandards organizations throughout the world. The goal of 3GPP is tomake a globally applicable third generation (3G) mobile phone systemspecification within the scope of the IMT-2000 (International MobileTelecommunications-2000) standard as defined by the InternationalTelecommunication Union. The 3GPP Long Term Evolution (“LTE”) Committeeis considering OFDM as well as OFDM/OQAM (Orthogonal Frequency DivisionMultiplexing/Offset Quadrature Amplitude Modulation), as a method fordownlink transmission, as well as OFDM transmission on the uplink.

Wireless communications systems (e.g., Time Division Multiple Access(TDMA), Orthogonal Frequency-Division Multiplexing (OFDM)) usuallycalculate an estimation of a channel impulse response between theantennas of a user device and the antennas of a base station forcoherent receiving. Channel estimation may involve transmitting knownreference signals that are multiplexed with the data. Reference signalsmay include a single frequency and are transmitted over thecommunication systems for supervisory, control, equalization,continuity, synchronization, etc. Wireless communication systems mayinclude one or more mobile stations and one or more base stations thateach transmit a reference signal. Reference signals may be designed suchthat an unacceptable peak cross-correlation between reference signalsexists. As such, benefits may be realized from systems and methods thatimprove the design of reference signals for spatially multiplexedcellular systems. In particular, benefits may be realized from systemsand methods that reduce peak to average cross-correlation for sequencesdesigned by alternating projections.

BRIEF DESCRIPTION OF THE DRAWINGS

Exemplary embodiments of the invention will become more fully apparentfrom the following description and appended claims, taken in conjunctionwith the accompanying drawings. Understanding that these drawings depictonly exemplary embodiments and are, therefore, not to be consideredlimiting of the invention's scope, the exemplary embodiments of theinvention will be described with additional specificity and detailthrough use of the accompanying drawings in which:

FIG. 1 illustrates an exemplary wireless communication system in whichembodiments may be practiced;

FIG. 2 illustrates some characteristics of a transmission band of an RFcommunication channel in accordance with an OFDM-based system;

FIG. 3 illustrates communication channels that may exist between an OFDMtransmitter and an OFDM receiver according to an embodiment;

FIG. 4 illustrates one embodiment of a MIMO system that may beimplemented with the present systems and methods;

FIG. 5 illustrates a block diagram of certain components in anembodiment of a transmitter;

FIG. 6 is a block diagram illustrating one embodiment of components usedto design a reference signal to be transmitted in a MIMO system;

FIG. 7 is a flow diagram illustrating one embodiment of a method fordesigning a reference signal in a MIMO system;

FIG. 8 is a flow diagram illustrating a further embodiment of analgorithm that may be utilized to design a reference signal;

FIG. 9 is a flow diagram illustrating a method of an algorithm that maybe utilized to design a reference signal in a MIMO system; and

FIG. 10 illustrates various components that may be utilized in acommunications device.

DETAILED DESCRIPTION

A method for using a numerical method to design reference signals formultiple input multiple output (MIMO) systems is described. An inputmultiple input multiple output signal is determined. A nearest tightframe to one or more given structured vectors is obtained. One or morestructured vectors from the nearest tight frame are obtained. Orthogonalsubsets are computed for a plurality of sequences, wherein each of thesubsets is replaced with a matrix that comprises a function of one ormore elements in each of the plurality of sequences and an identitymatrix associated with the one or more elements. The one or morestructured vectors are projected onto the space of circulant matrices.One or more classes of matrices associated with reference signals areoutputted.

In one embodiment, a set of reference signals is provided to cover threesectors of a cell. At least two reference signals per sector may beprovided. The set of reference signals may be orthogonal in each sectorof a given cell. The set of reference signals is orthogonal in sectorsadjacent to a given cell. In one embodiment, reference signals not inadjacent sectors are minimally correlated. The set of reference signalsmay include a Peak to Average Power Ratio that approximates the value ofone. Multiple bandwidths may be employed simultaneously. A set ofsequences may be recursively generated from a base sequence.

A matrix on the unit hyper-sphere with non-zero components may beprovided. The correlation between each of the one or more structuredvectors may be outputted. A sequence set may be projected to a nearesttight frame. Subsets of a nearest tight frame may be projected to one ormore orthogonal matrices. One or more orthogonal matrices may beprojected to a nearest circulant matrix. Each sequence may be projectedonto a minimum Peak to Average Power Ratio vector. The matrix may equal(βI_(d)+(1−β))(X_(i) X_(i) ^(H))^(1/2) X_(i), wherein d comprises thenumber of elements in any given sequence, I_(d) comprises a d×d identitymatrix, and 0≦β≦1.

A transmitter that is configured to use a numerical method to designreference signals for multiple input multiple output (MIMO) systems isalso described. The transmitter includes a processor and memory inelectronic communication with the processor. Instructions are stored inthe memory. An input multiple input multiple output signal isdetermined. A nearest tight frame to one or more given structuredvectors is obtained. One or more structured vectors from the nearesttight frame are obtained. Orthogonal subsets are computed for aplurality of sequences, wherein each of the subsets is replaced with amatrix that comprises a function of one or more elements in each of theplurality of sequences and an identity matrix associated with the one ormore elements. The one or more structured vectors are projected onto thespace of circulant matrices. One or more classes of matrices associatedwith reference signals are outputted.

A computer-readable medium comprising executable instructions for usinga numerical method to design reference signals for multiple inputmultiple output (MIMO) systems is also described. An input multipleinput multiple output signal is determined. A nearest tight frame to oneor more given structured vectors is obtained. One or more structuredvectors from the nearest tight frame are obtained. Orthogonal subsetsare computed for a plurality of sequences, wherein each of the subsetsis replaced with a matrix that comprises a function of one or moreelements in each of the plurality of sequences and an identity matrixassociated with the one or more elements. The one or more structuredvectors are projected onto the space of circulant matrices. One or moreclasses of matrices associated with reference signals are outputted.

Various embodiments of the invention are now described with reference tothe Figures, where like reference numbers indicate identical orfunctionally similar elements. The embodiments of the present invention,as generally described and illustrated in the Figures herein, could bearranged and designed in a wide variety of different configurations.Thus, the following more detailed description of several exemplaryembodiments of the present invention, as represented in the Figures, isnot intended to limit the scope of the invention, as claimed, but ismerely representative of the embodiments of the invention.

The word “exemplary” is used exclusively herein to mean “serving as anexample, instance, or illustration.” Any embodiment described herein as“exemplary” is not necessarily to be construed as preferred oradvantageous over other embodiments.

Many features of the embodiments disclosed herein may be implemented ascomputer software, electronic hardware, or combinations of both. Toclearly illustrate this interchangeability of hardware and software,various components will be described generally in terms of theirfunctionality. Whether such functionality is implemented as hardware orsoftware depends upon the particular application and design constraintsimposed on the overall system. Skilled artisans may implement thedescribed functionality in varying ways for each particular application,but such implementation decisions should not be interpreted as causing adeparture from the scope of the present invention.

Where the described functionality is implemented as computer software,such software may include any type of computer instruction or computerexecutable code located within a memory device and/or transmitted aselectronic signals over a system bus or network. Software thatimplements the functionality associated with components described hereinmay comprise a single instruction, or many instructions, and may bedistributed over several different code segments, among differentprograms, and across several memory devices.

As used herein, the terms “an embodiment”, “embodiment”, “embodiments”,“the embodiment”, “the embodiments”, “one or more embodiments”, “someembodiments”, “certain embodiments”, “one embodiment”, “anotherembodiment” and the like mean “one or more (but not necessarily all)embodiments of the disclosed invention(s)”, unless expressly specifiedotherwise.

The term “determining” (and grammatical variants thereof) is used in anextremely broad sense. The term “determining” encompasses a wide varietyof actions and therefore “determining” can include calculating,computing, processing, deriving, investigating, looking up (e.g.,looking up in a table, a database or another data structure),ascertaining and the like. Also, “determining” can include receiving(e.g., receiving information), accessing (e.g., accessing data in amemory) and the like. Also, “determining” can include resolving,selecting, choosing, establishing and the like.

The phrase “based on” does not mean “based only on,” unless expresslyspecified otherwise. In other words, the phrase “based on” describesboth “based only on” and “based at least on.”

Reference signals may be used in communication systems. Referencesignals may include a single frequency and are transmitted over thecommunication systems for supervisory, control, equalization,continuity, synchronization, etc. Communication systems may include oneor more mobile stations and one or more base stations that each transmita reference signal. Reference signals may be designed such that a mobilestation may re-use a reference signal that was previously used by adifferent mobile station or is used at the same time at another mobilestation in another cell far enough apart so as to negligibly interfere.Truncation or cyclic extension of a particular set of Zadoff-Chusequences has been utilized to design reference signals for re-use.However, truncation or cyclic extension may result in a tedious integerprogramming problem for a sequence assignment. In addition, a guaranteeof minimal correlation does not exist when truncation or cyclicextension of a particular set of Zadoff-Chu sequences are implemented.Further, because of variable correlation properties of candidateproposed reference signals, detailed planning regarding the mobilestation may be done which may be particularly vexing if adjacentnetworks exist in the same band that are operated by differentoperators. In addition, the designed reference signal sequences mayinclude an unacceptable peak cross-correlation.

The present systems and methods design reference signals for multipleinput multiple output (MIMO) systems in which reference signals areallocated amongst one or more mobile stations, for use in single user ormultiple user MIMO systems. In one embodiment, the present systems andmethods design uplink reference signals in a cellular system.Communications from mobile stations to base stations may be classifiedas “uplink” communications. Conversely, communications from basestations to mobile stations may be classified as “downlink”communications. Transmitting uplink reference signals in a cellularsystem may pose stringent requirements on time and frequency resourceson the mobile station. These stringent requirements may impede anoptimum design of the reference signals for the mobile station, whichmay desire to implement a single or multiple carrier modulation withcyclic prefix, where there is synchronization between the transmissionof multiple uplink signals and their respective base stations and wheresectorization amongst cells of mobile stations is employed to maximizethe capacity per cell. In addition, the present systems and methodsemploy multiple bandwidth allocations simultaneously to multiple basestations. In one embodiment, each bandwidth segment allocated to amobile station is an integer amount of some basic unit.

In designing a set of reference signals, certain design considerationsmay be implemented. For example, the set may be large enough to cover atleast three sectors per cell, with at least two reference signals persector. In one embodiment, four reference signals per sector arepresent. A further design consideration may be that the set of referencesignals may be orthogonal in each sector of a given cell. The set ofreference signals may also be orthogonal in all sectors adjacent to agiven sector. If the reference signals are orthogonal and the referencesignals are known to adjacent sectors, a best minimum mean squarereceiver may be designed and implemented.

For those reference signals that are not in adjacent sectors, or whichare not orthogonal, another design consideration may be that thesereference signal are minimally correlated, with approximately the samecorrelation, and approach (if not meet) the Welch Bound. Sets ofsequences that approach or meet the Welch Bound may denote a tightframe, where each vector possesses a unit norm, i.e., ∥χη∥₂=1. A furtherdesign consideration is the set of reference signals may also have aPeak to Average Power Ratio (PAPR) that approaches (if not equal) to 1.The PAPR may be defined as, for a sequence vector c as:

$\begin{matrix}{{{??} = \frac{{c}_{\infty}^{2}}{c^{H}c}},} & \left( {{Equation}\mspace{20mu} 1} \right)\end{matrix}$where ∥c∥_(∞) ² denotes the square maximum modulus component of c andwhere ( )^(H) denotes a conjugate transpose.

Another example of a design consideration may be that amongst subsets ofsequences with orthogonal elements, each element may be a cyclic shiftof another element. This property may be useful to provide robustperformance if a transmission system which transmits a cyclic prefix formultipath elimination encounters multipath components with a delayspread greater than the cyclic prefix length. An additional designconsideration is that in a system where multiple bandwidths are employedsimultaneously, the set of reference signal sequences may be recursivelygenerated from a base sequence.

In one embodiment, the amount of reference signal space (time andfrequency resources) may be exactly large enough. For example, the basicunit of bandwidth allocation may allow for 19 or any larger prime numberof reference signals available for two reference signals per sector. Ina further example, the basic unit of bandwidth allocation may allow for37 or any larger prime number of reference signals for four referencesignals per sector. As in this case, if the amount of reference signalspace is exactly large enough, Zadoff-Chu sequences may be taken as thereference sequences as they meet the design considerations previouslydescribed. However, such resource availability or sequence numerologymay not be plausible. An algorithm may be implemented for designingreference signals based on alternating projections. The algorithm mayinclude a series of linear transformation that result in referencesignal sequences that are nearest in Euclidean space to a “tight frame,”(a set of minimally correlation sequences) as well as transformationsthat result in orthogonal subsets of sequences and cyclic shifts ofsequences. However, the transformation that results in orthogonalsubsets of sequences may often create sets of sequences that have highlycorrelation sequences, despite the overall minimization of meancross-correlation. For example, previous systems and methods providethat subsets of a sequence X_(i), should be replaced with Y_(i)=(X_(i)X_(i) ^(H))^(1/2) X_(i). However, this transformation may undulyincrease the maximum cross-correlation of sequences. In other words,Y_(i) ^(H) Y_(j) may have elements whose magnitude is greater than ½ andclose to unity The present systems and methods reduce peak to averagecross-correlation for sequences designed by alternating projections.

FIG. 1 illustrates an exemplary wireless communication system 100 inwhich embodiments may be practiced. A base station 102 is in wirelesscommunication with a plurality of user devices 104 (which may also bereferred to as mobile stations, subscriber units, access terminals,etc.). A first user device 104 a, a second user device 104 b, and an Nthuser device 104 n are shown in FIG. 1. The base station 102 transmitsdata to the user devices 104 over a radio frequency (RF) communicationchannel 106.

As used herein, the term “OFDM transmitter” refers to any component ordevice that transmits OFDM signals. An OFDM transmitter may beimplemented in a base station 102 that transmits OFDM signals to one ormore user devices 104. Alternatively, an OFDM transmitter may beimplemented in a user device 104 that transmits OFDM signals to one ormore base stations 102.

The term “OFDM receiver” refers to any component or device that receivesOFDM signals. An OFDM receiver may be implemented in a user device 104that receives OFDM signals from one or more base stations 102.Alternatively, an OFDM receiver may be implemented in a base station 102that receives OFDM signals from one or more user devices 104.

FIG. 2 illustrates some characteristics of a transmission band 208 of anRF communication channel 206 in accordance with an OFDM-based system. Asshown, the transmission band 208 may be divided into a number of equallyspaced sub-bands 210. As mentioned above, a sub-carrier carrying aportion of the user information is transmitted in each sub-band 210, andevery sub-carrier is orthogonal with every other sub-carrier.

FIG. 3 illustrates communication channels 306 that may exist between anOFDM transmitter 312 and an OFDM receiver 314 according to anembodiment. As shown, communication from the OFDM transmitter 312 to theOFDM receiver 314 may occur over a first communication channel 306 a.Communication from the OFDM receiver 314 to the OFDM transmitter 312 mayoccur over a second communication channel 306 b.

The first communication channel 306 a and the second communicationchannel 306 b may be separate communication channels 306. For example,there may be no overlap between the transmission band of the firstcommunication channel 306 a and the transmission band of the secondcommunication channel 306 b.

In addition, the present systems and methods may be implemented with anymodulation that utilizes multiple antennas/MIMO transmissions. Forexample, the present systems and methods may be implemented for MIMOCode Division Multiple Access (CDMA) systems or Time Division MultipleAccess (TDMA) systems.

FIG. 4 illustrates one embodiment of a MIMO system 400 that may beimplemented with the present systems and methods. The illustrated MIMOsystem 400 includes a first transmit antenna (Tx₁) 402A and a secondtransmit antenna (Tx₂) 402B. The system 400 also includes a firstreceive antenna (Rx₁) 404A and a second receive antenna (Rx₂) 404B. Thetransmit antennas 402A, 402B may be used to transmit a signal 406, 408,410, 412 to the receive antennas 404A, 404B.

In single antenna systems, multi-path propagation may be detrimental tothe performance of the system. The multiple propagation paths may cause“copies” of a signal to arrive at a receiver at slightly differenttimes. These time delayed signals may then become interference whentrying to recover the signal of interest. The MIMO system 400 isdesigned to exploit the multi-path propagation to obtain a performanceimprovement. For example, the first receive antenna (Rx₁) 404A mayreceive a mixture of a first signal 406 and a third signal 410 which aresent from the first transmit antenna (Tx₁) 402A and the second transmitantenna (Tx₂) 402B. The first and third signals 406, 410 may be sentover a first channel h_(1,1) and a second third channel h_(2,1). Theproportion of the first and third signals that is received at the firstreceive antenna (Rx₁) 404A depends on the transmission channels h_(1,1),h_(2,1). A simplified equation for the signal received at the firstreceive antenna (Rx₁) 404A may be:Rx ₁=(h _(1,1) ×Tx ₁)+(h _(2,1) ×Tx ₂)  (Equation 2)

The first receive antenna (Rx₁) 404A receives a combination of what wastransmitted from the first and second transmit antennas 402A, 402B. TheMIMO system 400 may implement various coding schemes that define whichsignals 406, 408, 410, 412 should be transmitted, and at what times, toenable an original signal to be recovered when it is received incombination with another signal. These coding schemes may be known as“space-time” codes because they define a code across space (antennas)and time (symbols).

FIG. 5 illustrates a block diagram 500 of certain components in anembodiment of a transmitter 504. Other components that are typicallyincluded in the transmitter 504 may not be illustrated for the purposeof focusing on the novel features of the embodiments herein.

Data symbols may be modulated by a modulation component 514. Themodulated data symbols may be analyzed by other subsystems 518. Theanalyzed data symbols 516 may be provided to a reference processingcomponent 510. The reference processing component 510 may generate areference signal that may be transmitted with the data symbols. Themodulated data symbols 512 and the reference signal 508 may becommunicated to an end processing component 506. The end processingcomponent 506 may combine the reference signal 508 and the modulateddata symbols 512 into a signal. The transmitter 504 may receive thesignal and transmit the signal to a receiver through an antenna 502.

FIG. 6 is a block diagram 600 illustrating one embodiment of componentsused to design a reference signal to be transmitted in a MIMO system. Inone embodiment, an initial sequence retriever 602 may obtain initialsequences. A first sequence projection component 604 may project anobtained sequence set to a nearest tight frame. A subsets projectioncomponent 606 may be implemented to project subsets of the nearest tightframe to one or more orthogonal matrices. In one embodiment, a matricesprojection component 608 may project the one or more orthogonal matricesto a nearest circulant matrix. In one embodiment, a second sequenceprojection component 610 may project each of the obtained sequence setsonto a minimum Peak to Average Power Ratio (PAPR) vector. An iterator612 may be utilized to iterate the steps performed by the first sequenceprojection component 604, the subsets projection component 606, thematrices projection component 608 and the second sequence projectioncomponent 610. The iterator 612 may iterate these steps T times. Asequence output component 614 may output the sequences after Titerations have been executed.

FIG. 7 is a flow diagram illustrating one embodiment of a method 700 fordesigning a reference signal in a MIMO system. The method 700 may beimplemented by the components discussed previously in regards to FIG. 6.In one embodiment, the existence of a fixed point of a MIMO signal isverified 702. For example, for a set of Zadoff-Chu sequences of lengths19 or 37 (as previously described), the Zadoff-Chu sequences may bereturned as used as an input to design the reference signal. A nearesttight frame to one or more structured vectors may be obtained 704. Oneor more structured vectors may then be obtained 706 from the previouslycomputed nearest tight frame. The one or more structured vectors may beprojected 708 onto the space of circulant matrices and one or moreclasses of matrices may be outputted 710. The outputted matrices mayindicate the design of the reference signal transmitted in a MIMOsystem.

FIG. 8 is a flow diagram 800 illustrating a further embodiment of analgorithm that may be utilized to design a reference signal. In oneembodiment, a first matrix is provided 802. The first matrix may be onthe unit hyper-sphere. Sequences may be on the unit hyper-sphere toensure a satisfactory constant envelope property initially. The firstmatrix may include zero components if the starting sequence is on theunit hyper-sphere. A second matrix may be computed 804. The secondmatrix may be a nearest tight frame to the first matrix. The nearesttight frame may include an estimation of the first matrix.

In one embodiment, a third matrix may be computed 806. The third matrixmay be the closest matrix with a minimum peak to average power ratio tothe second matrix. The third matrix may also be expanded and a fourthmatrix may be computed 808 from the expansion. In one embodiment, afifth matrix is computed 810 that is a nearest circulant matrix to thefourth matrix. The first matrix may be set 812 to the fifth matrix. Inother words, the first matrix may be assigned the included in the fifthmatrix. The fourth matrix and the fifth matrix may be outputted 814. Inaddition, a maximum inner product of the fourth and fifth matrices mayalso be outputted 814.

The following may represent steps taken to compute a correlated set ofmatrices that is the closest matrix with a minimum peak to average powerratio. A sequence of N column vectors {x_(n)}_(n=1) ^(N), x_(n)ε

^(d), d≦N, may be assigned as columns of a matrix X=[x₁ x₂ . . . x_(N)].The matrix may be referred to as a frame. Each vector may have unitlength, without any loss in generality. Block of K of these vectors maybe grouped into a set of matrices, {X_(i)}_(i=1) ^(K) so that (withMK=N) X=[X₁ X₂ . . . X_(M)]. The correlation between vectors may berepresented as <x_(k), x_(n)> which is the standard inner product incomplex Euclidean d-space.

The Welch Bound is, for any frame, for k≠n:

$\begin{matrix}{{\max_{k \neq n}{\text{<}x_{k}}},{{x_{n}\text{>}} \geq \sqrt{\frac{N - d}{d\left( {N - 1} \right)}}}} & \left( {{Equation}\mspace{20mu} 3} \right)\end{matrix}$

A frame that meets or approaches the Welch Bound may be referred to as atight frame. The design considerations previously mentioned imply thatfor any <x_(k), x_(n)> not in the same X_(i), <x_(k), x_(n)>≦α, where αis a constant determined by the Welch Bound provided above. If anymatrix Zε

^(d×N), is provided, the matrix that comes closest in distance (asmeasured in element-wise or Frobenius norm) may be given by α(ZZ^(H))^(1/2) Z. This condition may also enforce an orthnormalitycondition between rows of X, if an optimal X exists.

The design considerations previously mentioned also imply thatX_(i)*X_(i)=I_(K)*; (with K≦d). In other words, each column in any X_(i)may be orthogonal to any other column in X_(i). The above may berepeated with the role of X above being assumed by X_(i) ^(H). Further,if as few as two sequences are required per cell (i.e., per matrixX_(i)), a “phase parity check” may be implemented to provideorthogonality between column vectors in X_(i) when there are zeroentries in any column of X_(i). In other words, the phase of the zerocomponents are chosen such that orthogonality if maintained once eachcolumn vector has minimal Peak to Average Power Ratio.

The following may illustrate steps taken to obtain the circulant matrixnearest to a given matrix. A matrix Z=[z₁ . . . z_(N)], may be provided,where each z_(i) is a column vector ε

^(N). A circulant matrix C=[c₀ . . . c_(N−1)], may be obtained that isclosest in Frobenius (element-wise) norm to Z. In one embodiment, F maybe given as the Discrete Fourier Transform (DFT) matrix:

$\begin{matrix}{F = \begin{bmatrix}1 & 1 & \cdots & 1 \\1 & {\mathbb{e}}^{{- j}\; 2{\pi/N}} & \cdots & {\mathbb{e}}^{{- j}\; 2{{\pi{({N - 1})}}/N}} \\\vdots & \vdots & \ddots & \vdots \\1 & {\mathbb{e}}^{{- {{j2\pi}{({N - 1})}}}/N} & \cdots & {\mathbb{e}}^{{- {{j2\pi}{({N - 1})}}}{{({N - 1})}/N}}\end{bmatrix}} & \left( {{Equation}\mspace{20mu} 4} \right)\end{matrix}$

A diagonal “delay” matrix D may be defined as D=diag(1 e^(−j2π/N)e^(−j2π2/N) . . . e^(−j2π(N−1)/N)). For any ciculant matrix C,C=F^(H)ΛF, where Λ is the DFT of the sequence/vector c₀. In addition, itmay be shown that c_(i+1 mod N)=F^(H)DF c_(i)=(F^(H)DF)^((i+1) mod N)c₀. Then

$\begin{matrix}{{{Z - C}}_{F}^{2} = {\sum\limits_{i = 1}^{N}\;{{z_{i} - c_{i - 1}}}^{2}}} \\{= {\sum\limits_{i = 1}^{N}\;{{{z_{i} - {\left( {F^{H}D\; F} \right)^{({i - 1})}c_{0}}}}^{2}.}}}\end{matrix}$

In one embodiment,

${\zeta = \begin{bmatrix}z_{1} \\z_{2} \\\vdots \\z_{N}\end{bmatrix}},\mspace{14mu}{{{and}\mspace{14mu}} = \begin{bmatrix}I_{N} \\{F^{H}D\; F} \\\vdots \\\left( {F^{H}D\; F} \right)^{N - 1}\end{bmatrix}}$to minimize c₀, which uniquely determines C, c₀ is given by c₀=

⁺ζ, where

⁺ is the Moore-Penrose pseudo-inverse of

. In other words,

⁺=

^(H)

⁻¹

^(H).

Matrices where the number of column vectors are not equal to the numberof row vectors may be referred to as reduced rank matrices (Z has fewerthan N columns). Modifications may be implemented to the recurrencerelation c_(i+1 mod N)=F^(H)DF c_(i) and the forming of the appropriatematrix

. If only two vectors were required that were cyclic shifted threeelements apart, then c₁=(F^(H)DF)³ c₀ and

may include the matrix elements I_(N) and (F^(H)DF)².

FIG. 9 is a flow diagram 900 illustrating a method of an algorithm thatmay be utilized to design a reference signal in a MIMO system. A matrixZ₀ε

^(d×N), may be provided 902. In one embodiment, the matrix Z₀ is on theunit hyper-sphere with all non-zero components. The following may occurfor t=1 to T.

Orthogonal subsets for a plurality of sequences may also be created.Each subset of sequence may be replaced with matrix that is equal to(βI_(d)+(1−β))(z_(i) z_(i) ^(H))^(1/2) z_(i). For example,(βI_(d)+(1−β))(z_(i) z_(i) ^(H))^(1/2) z_(i) may be computed 904 andassigned to the matrix Y. This may result in the tight frame nearest toZ. The value d may represent the number of elements in a given sequence,I_(d) may be a d×d identity matrix and 0≦β≦1. In one embodiment, β≅0.1may provide a satisfactory tradeoff between maximum correlation andorthogonality. The following constraints may be implemented. If zeroentries exist in column vectors of Y, phases to their related componentsin Y may be added so that orthogonality is maintained. For m=1 to M,(α(W_(m) ^(H) W_(m))^(1/2) W_(m) ^(H))^(H) may be computed 906 andassigned to a vector V_(m). The matrix V=[V₁ V₂ . . . V_(M)] may beassembled.

In one embodiment, the max_(k≠n)<ν_(k), ν_(n)> may be computed. Further,a Q matrix may be computed 908 that is a nearest circulant matrix to Vand max_(k≠n)<q_(k), q_(n)> may also be computed. A W matrix may becomputed 910. The W matrix may be the closest matrix with minimum PAPRto Y. The W matrix may be expressed as W=[W₁ W₂ . . . W_(M)]. The Zmatrix may be assigned 912 as the Q matrix. If a circulant matrix is notdesired, the Z matrix may be assigned as the V matrix. In oneembodiment, t is updated as t+1. The V matrix and the Q matrix may beoutputted 914. In addition, max_(k≠n)<ν_(k), ν_(n)> and max_(k≠n)<q_(k),q_(n)> may also be outputted 914.

FIG. 10 illustrates various components that may be utilized in acommunications device 1002. The communications device 1002 may includeany type of communications device such as a mobile station, a cellphone, an access terminal, user equipment, a base station transceiver, abase station controller, etc. The communications device 1002 includes aprocessor 1006 which controls operation of the communications device1002. The processor 1006 may also be referred to as a CPU. Memory 1008,which may include both read-only memory (ROM) and random access memory(RAM), provides instructions and data to the processor 1006. A portionof the memory 1008 may also include non-volatile random access memory(NVRAM).

The communications device 1002 may also include a housing 1022 thatcontains a transmitter 1012 and a receiver 1014 to allow transmissionand reception of data. The transmitter 1012 and receiver 1014 may becombined into a transceiver 1024. An antenna 1026 is attached to thehousing 1022 and electrically coupled to the transceiver 1024.Additional antennas (not shown) may also be used.

The communications device 1002 may also include a signal detector 1010used to detect and quantify the level of signals received by thetransceiver 1024. The signal detector 1010 detects such signals as totalenergy, pilot energy, power spectral density, and other signals.

A state changer 1016 controls the state of the communications device1002 based on a current state and additional signals received by thetransceiver 1024 and detected by the signal detector 1010. Thecommunications device 1002 may be capable of operating in any one of anumber of states.

The various components of the communications device 1002 are coupledtogether by a bus system 1020 which may include a power bus, a controlsignal bus, and a status signal bus in addition to a data bus. However,for the sake of clarity, the various buses are illustrated in FIG. 10 asthe bus system 1020. The communications device 1002 may also include adigital signal processor (DSP) 1018 for use in processing signals. Thecommunications device 1002 illustrated in FIG. 10 is a functional blockdiagram rather than a listing of specific components.

Information and signals may be represented using any of a variety ofdifferent technologies and techniques. For example, data, instructions,commands, information, signals, bits, symbols, and chips that may bereferenced throughout the above description may be represented byvoltages, currents, electromagnetic waves, magnetic fields or particles,optical fields or particles, or any combination thereof.

The various illustrative logical blocks, modules, circuits, andalgorithm steps described in connection with the embodiments disclosedherein may be implemented as electronic hardware, computer software, orcombinations of both. To clearly illustrate this interchangeability ofhardware and software, various illustrative components, blocks, modules,circuits, and steps have been described above generally in terms oftheir functionality. Whether such functionality is implemented ashardware or software depends upon the particular application and designconstraints imposed on the overall system. Skilled artisans mayimplement the described functionality in varying ways for eachparticular application, but such implementation decisions should not beinterpreted as causing a departure from the scope of the presentinvention.

The various illustrative logical blocks, modules, and circuits describedin connection with the embodiments disclosed herein may be implementedor performed with a general purpose processor, a digital signalprocessor (DSP), an application specific integrated circuit (ASIC), afield programmable gate array signal (FPGA) or other programmable logicdevice, discrete gate or transistor logic, discrete hardware components,or any combination thereof designed to perform the functions describedherein. A general purpose processor may be a microprocessor, but in thealternative, the processor may be any conventional processor,controller, microcontroller, or state machine. A processor may also beimplemented as a combination of computing devices, e.g., a combinationof a DSP and a microprocessor, a plurality of microprocessors, one ormore microprocessors in conjunction with a DSP core, or any other suchconfiguration.

The steps of a method or algorithm described in connection with theembodiments disclosed herein may be embodied directly in hardware, in asoftware module executed by a processor, or in a combination of the two.A software module may reside in RAM memory, flash memory, ROM memory,EPROM memory, EEPROM memory, registers, hard disk, a removable disk, aCD-ROM, or any other form of storage medium known in the art. Anexemplary storage medium is coupled to the processor such that theprocessor can read information from, and write information to, thestorage medium. In the alternative, the storage medium may be integralto the processor. The processor and the storage medium may reside in anASIC. The ASIC may reside in a user terminal. In the alternative, theprocessor and the storage medium may reside as discrete components in auser terminal.

The methods disclosed herein comprise one or more steps or actions forachieving the described method. The method steps and/or actions may beinterchanged with one another without departing from the scope of thepresent invention. In other words, unless a specific order of steps oractions is required for proper operation of the embodiment, the orderand/or use of specific steps and/or actions may be modified withoutdeparting from the scope of the present invention.

While specific embodiments and applications of the present inventionhave been illustrated and described, it is to be understood that theinvention is not limited to the precise configuration and componentsdisclosed herein. Various modifications, changes, and variations whichwill be apparent to those skilled in the art may be made in thearrangement, operation, and details of the methods and systems of thepresent invention disclosed herein without departing from the spirit andscope of the invention.

1. A method for using a numerical method to design reference signals formultiple input multiple output (MIMO) systems, the method comprising:determining an input multiple input multiple output signal; obtaining anearest tight frame to one or more given structured vectors, whereineach tight frame meets or approaches a Welch Bound; obtaining one ormore structured vectors from the nearest tight frame; computingorthogonal subsets for a plurality of sequences, wherein each of thesubsets is replaced with a matrix that comprises a function of one ormore elements in each of the plurality of sequences and a product of anidentity matrix associated with the one or more elements and a value,the value being greater than or equal to zero and less than or equal toone, wherein one less the value is added to the product; projecting theone or more structured vectors onto a space of circulant matrices; andoutputting one or more classes of matrices associated with referencesignals, wherein the method is performed by a communications device. 2.The method of claim 1, further comprising providing a set of referencesignals to cover three sectors of a cell.
 3. The method of claim 2,wherein at least two reference signals per sector are provided.
 4. Themethod of claim 2, wherein the set of reference signals is orthogonal ineach sector of a given cell.
 5. The method of claim 2, wherein the setof reference signals is orthogonal in sectors adjacent to a given cell.6. The method of claim 2, further comprising minimally correlatingreference signals not in adjacent sectors.
 7. The method of claim 2,wherein the set of reference signals comprise a Peak to Average PowerRatio that approximates a value of one.
 8. The method of claim 1,wherein multiple bandwidths are employed simultaneously.
 9. The methodof claim 8, wherein a set of sequences are recursively generated from abase sequence.
 10. The method of claim 1, further comprising providing amatrix on a unit hyper-sphere with non-zero components.
 11. The methodof claim 1, further comprising outputting the correlation between eachof the one or more structured vectors.
 12. The method of claim 1,further comprising projecting a sequence set to a nearest tight frame.13. The method of claim 1, further comprising projecting subsets of anearest tight frame to one or more orthogonal matrices.
 14. The methodof claim 1, further comprising projecting one or more orthogonalmatrices to a nearest circulant matrix.
 15. The method of claim 1,further comprising projecting each sequence onto a minimum Peak toAverage Power Ratio vector.
 16. The method of claim 1, wherein thematrix equals (βI_(d)+(1−β))(x_(i) x_(i) ^(H))^(1/2) x_(i), wherein dcomprises the number of elements in any given sequence, I_(d) comprisesa d×d identity matrix, 0≦β≦1, X_(i) comprises the one or more elementsin each of the plurality of sequences and H denotes a conjugatetranspose.
 17. A transmitter that is configured to use a numericalmethod to design reference signals for multiple input multiple output(MIMO) systems, the transmitter comprising: a processor; memory inelectronic communication with the processor; instructions stored in thememory, the instructions being executable to: determine an inputmultiple input multiple output signal; obtain a nearest tight frame toone or more given structured vectors, wherein each tight frame meets orapproaches a Welch Bound; obtain one or more structured vectors from thenearest tight frame; compute orthogonal subsets for a plurality ofsequences, wherein each of the subsets is replaced with a matrix thatcomprises a function of one or more elements in each of the plurality ofsequences and a product of an identity matrix associated with the one ormore elements and a value, the value being greater than or equal to zeroand less than or equal to one, wherein one less the value is added tothe product; project the one or more structured vectors onto a space ofcirculant matrices; and output one or more classes of matricesassociated with reference signals.
 18. The transmitter of claim 17,wherein the instructions are further executable to provide a matrix on aunit hyper-sphere with non-zero components.
 19. The transmitter of claim17, wherein the instructions are further executable to output thecorrelation between each of the one or more structured vectors.
 20. Thetransmitter of claim 17, wherein the instructions are further executableto provide a set of reference signals to cover three sectors of a cell.21. The transmitter of claim 20, wherein at least two reference signalsper sector are provided.
 22. The transmitter of claim 20, wherein theset of reference signals is orthogonal in each sector of a given cell.23. The transmitter of claim 20, wherein the set of reference signals isorthogonal in sectors adjacent to a given cell.
 24. The transmitter ofclaim 20, wherein the instructions are further executable to minimallycorrelate reference signals not in adjacent sectors.
 25. Anon-transitory computer-readable medium comprising executableinstructions for using a numerical method to design reference signalsfor multiple input multiple output (MIMO) systems, the instructionsbeing executable to: determine an input multiple input multiple outputsignal; obtain a nearest tight frame to one or more given structuredvectors, wherein each tight frame meets or approaches a Welch Bound;obtain one or more structured vectors from the nearest tight frame;compute orthogonal subsets for a plurality of sequences, wherein each ofthe subsets is replaced with a matrix that comprises a function of oneor more elements in each of the plurality of sequences and a product ofan identity matrix associated with the one or more elements and a value,the value being greater than or equal to zero and less than or equal toone, wherein one less the value is added to the product; project the oneor more structured vectors onto a space of circulant matrices; andoutput one or more classes of matrices associated with referencesignals.